A new O(N) method for modeling and simulating the behavior of a large number of dislocations in anisotropic linear elastic media

An O(N) method is developed for calculating the interactions of a large number, N, of parallel straight dislocations in a solid whose linear elastic properties exhibit general anisotropy. The effect of anisotropy is accounted for by using a dislocation-dislocation interaction energy derived from the Stroh formalism for plane elastostatics. For simulations allowing the use of doubly periodic boundary conditions the interaction energy between an infinite dislocation wall and a single dislocation under conditions of general anisotropy is of more utility and has been developed by us; in combination with a fast multipole method (valid for anisotropy) and one-dimensional space partitioning, the extremely short-range nature of the dislocation-dislocation wall interaction force leads to a tremendously efficient and accurate computational scheme for determining equilibrium dislocation configurations as well as the kinetic evolution (via a non-linear velocity - Peach-Koehler force law) of dislocation motion in an O(N) scheme. The kinetic simulations also allow for dislocation generation and annihilation. As examples of the utility of the methods we present simulations of equilibrium dislocation arrangements, dislocation cell formation, slip band formation, and machine testing in single crystals. The first application is to seek equilibrium states of dislocations using the Conjugate Gradient (CG) method. The second application is a study of the dynamical evolution of dislocation patterns in copper single crystals with two active slip systems; these simulations can be carried far enough to provide evidence of dislocation (Mughrabi) cell formation. The third application shows slip band formation consistent with experimental observations. Other simulations in which the macroscopic strain rate is controlled using feedback methods allow us to generate the macroscopic stress-strain curve during both loading and unloading stages in a dynamic simulation involving over 20,000 dislocations in a basic computational cell (with periodic B.C.'s) and to track the total and mobile dislocation density as a function of strain. These numerical studies clearly indicate that such simulations provide a practical method for understanding and predicting dislocation pattern formation as well as post-yield behavior, including work-hardening.